Chris McDaniel, the tea-party backed Republican challenger who lost the Mississippi GOP runoff election to Sen. Thad Cochran, now says the election was “a sham, plain and simple,” and is raising money to mount a legal challenge.

“My opponent stole last week’s runoff election, and I’m not going down without a fight,” Mr. McDaniel said in recent fundraising appeal.

At issue is the outcome of the June 24 runoff, which Mr. Cochran won by 6,700 votes, reversing the outcome of the June 3 primary, which Mr. McDaniel won by about 1,400 votes in one of the nastiest, most-competitive GOP primaries in the country this year.

Mr. McDaniel has yet to concede the election and his campaign staff has been trying to scrutinize ballots because they suspect that some of the voters who supported Mr. Cochran were voting illegally. Under state law, people who voted in the June 3 Democratic primary are not allowed to vote in the June 24 Republican runoff. The McDaniel campaign says its volunteers have found 3,300 voting irregularities in 38 of Mississippi’s 82 counties.

Mr. McDaniel also is supporting a lawsuit filed Tuesday against the state of Mississippi by a conservative group, True the Vote, that is seeking access to election records and more time to review them for possible illegality.

In a statement supporting the group’s suit, Mr. McDaniel said in a written statement Wednesday, “The voters should be able to trust that the manner in which their elected officials are chosen is not compromised.”

Mr. McDaniel, in his fundraising letter seeking support for his own legal challenge, told supporters, “We have a long fight ahead of us. I know exactly how long and frustrating court battles can be, but I believe this will be worth it.”

The Cochran campaign dismissed the legal maneuvering as the response of a sore loser.

“Chris McDaniel is accusing the Governor of Mississippi, the Secretary of State and all elected officials of being incapable of holding free and fair elections,” said Stuart Stevens, a senior Cochran adviser. “His proof? He lost. Had he won by 100 votes, he would be celebrating.”

Austin Barbour, another senior adviser to the campaign, said the McDaniel camp exaggerated the number of ballot “irregularities.” In an interview Wednesday after a press conference call was disrupted by an unruly questioner, he said: “It’s time to move on because there is nothing there.”

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